Need some explanation of Pumping lemma for CFL

Question

I need some help with the understanding of Pumping Lemma for CFL

L = {all words over $\{a,b,c\}$ s.t. $n_a=n_b+2n_c\}$ where $n_a$ stands for number of $a$,$n_b$ - number of $b$ and $n_c$ number of $c$.
show that $L$ is not $CFL$

My thoughts:
Assume $L$ is $CFL$
take $w = b^{m}c^{2m}a^{3m}$.
$vy$ is either $b^{i} c^{j}$ or $c^{i} c^{j}$ or $c^{i} a^{j}$ or $a^{i} a^{j}$
take $w_1=uvvxyyz$
$w_1$ doesn't in $L$ since it doesn't satisfie the rule $n_a=n_b+2n_c $ Thus $L$ is not $CFL$

Can you show me where I'm wrong?

Answer 1

I hope it can help you.

Pumping lemma for CFL :

for infinite context-free language L

there exists an integer $m$ such that:

for any string $w\in L$ ,$|w| \ge m$

we can write $w=uvxy$

1. $|vxy|\le m$

2. $|vy| \ge 1$

and it must be: $\,\,\,\,\,\,uv^ixy^iz\in L$ $\,\,\,\,\,\,\,\,\forall i \ge 0 $

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The language $L$ is not context-free language

Use the Pumping Lemma for context-free languages .we proof by contradiction (assume L is CFL pick any string of L with length at least m) examine $\underline{all}$ the possible locations of string $vxy$ in string $w$ and show contradiction in each case .

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A language is context-free if there is a CFG for it

A context-free grammar (CFG) consists of a set of productions that you use to replace a variable by a string of variables and terminals. The language of a grammar is the set of strings it generates. A language is context-free if there is a CFG for it

your language in question is context-free language.

for $L=\{w\in\{a,b,c\}^*: n_a=n_b+2n_c , n_a \ge 0,n_b \ge 0,n_c \ge 0\}$ we have this grammar :

$\,\,\,\,\,\,\,\,\,S\to aaSc|A$
$\,\,\,\,\,\,\,\,\,A\to aAb|\epsilon$

so exist a context-free grammar that generating this language.

$\,\,\,\,\,\,\,\,$=> L is context-free language.